Octave 3.8, jcobi/3

Average Error: 3.6 → 0.1

Time: 23.9s

Precision: binary64

Cost: 1600

\[\alpha > -1 \land \beta > -1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \frac{\frac{1 + \alpha}{\frac{t_0}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{t_0} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (/ (/ (+ 1.0 alpha) (* (/ t_0 (+ 1.0 beta)) (+ alpha (+ beta 3.0)))) t_0)))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = beta + (alpha + 2.0d0)
    code = ((1.0d0 + alpha) / ((t_0 / (1.0d0 + beta)) * (alpha + (beta + 3.0d0)))) / t_0
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(t_0 / Float64(1.0 + beta)) * Float64(alpha + Float64(beta + 3.0)))) / t_0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 3.6

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

2. Simplified 2.0

   \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 3} \cdot \left(\beta + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
   Proof
   (/.f64 (*.f64 (/.f64 (+.f64 alpha 1) (+.f64 (+.f64 alpha beta) 3)) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (+.f64 (+.f64 alpha beta) 3)) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 24 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (+.f64 2 1)))) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) 1)) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) 1))) (+.f64 beta 1)) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) 1))) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 22 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (Rewrite=> /-rgt-identity_binary64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 21 points decrease in error
   (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (+.f64 beta 1))) (*.f64 (+.f64 alpha (+.f64 beta 2)) (+.f64 alpha (+.f64 beta 2)))): 10 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (+.f64 beta 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 11 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 alpha (+.f64 beta 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) (/.f64 (+.f64 beta 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta 1) (+.f64 1 alpha))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 1 alpha) (*.f64 beta (+.f64 1 alpha)))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 1 alpha) (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 beta 1) (*.f64 beta alpha)))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 1 alpha) (+.f64 (Rewrite=> *-rgt-identity_binary64 beta) (*.f64 beta alpha))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 alpha (+.f64 beta (*.f64 beta alpha))))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1 + \beta}{\beta + \left(\alpha + 2\right)}} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\frac{\beta + \left(2 + \alpha\right)}{\beta + 1} \cdot \left(\left(\beta + 3\right) + \alpha\right)}}{\beta + \left(2 + \alpha\right)}} \]
   Proof
   (/.f64 (/.f64 (+.f64 1 alpha) (*.f64 (/.f64 (+.f64 beta (+.f64 2 alpha)) (+.f64 beta 1)) (+.f64 (+.f64 beta 3) alpha))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha 1)) (*.f64 (/.f64 (+.f64 beta (+.f64 2 alpha)) (+.f64 beta 1)) (+.f64 (+.f64 beta 3) alpha))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 9 points decrease in error
   (/.f64 (/.f64 (+.f64 alpha 1) (*.f64 (/.f64 (+.f64 beta (Rewrite=> +-commutative_binary64 (+.f64 alpha 2))) (+.f64 beta 1)) (+.f64 (+.f64 beta 3) alpha))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 alpha 1) (*.f64 (/.f64 (+.f64 beta (+.f64 alpha 2)) (Rewrite=> +-commutative_binary64 (+.f64 1 beta))) (+.f64 (+.f64 beta 3) alpha))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 alpha 1) (*.f64 (/.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 1 beta)) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta 3))))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 beta 3))) (/.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 1 beta)))) (+.f64 beta (+.f64 2 alpha))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 beta 3))) (+.f64 beta (+.f64 alpha 2))) (+.f64 1 beta))) (+.f64 beta (+.f64 2 alpha))): 1 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (/.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 beta 3))) (+.f64 beta (+.f64 alpha 2))) (+.f64 1 beta)) (+.f64 beta (Rewrite=> +-commutative_binary64 (+.f64 alpha 2)))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (/.f64 (+.f64 alpha 1) (+.f64 alpha (+.f64 beta 3))) (+.f64 beta (+.f64 alpha 2))) (/.f64 (+.f64 1 beta) (+.f64 beta (+.f64 alpha 2))))): 9 points increase in error, 0 points decrease in error

5. Final simplification 0.1

   \[\leadsto \frac{\frac{1 + \alpha}{\frac{\beta + \left(\alpha + 2\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\beta + \left(\alpha + 2\right)} \]

Alternatives

Alternative 1
Error	0.1
Cost	1600

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \frac{\frac{-1 - \beta}{t_0} \cdot \frac{-1 - \alpha}{t_0}}{\alpha + \left(\beta + 3\right)} \end{array} \]

Alternative 2
Error	1.0
Cost	1472

\[\frac{\frac{1 + \alpha}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \frac{\beta + 2}{1 + \beta}}}{\beta + \left(\alpha + 2\right)} \]

Alternative 3
Error	0.8
Cost	1348

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 3}}{\beta + 2} \cdot \frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(4 + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1344

\[\frac{\frac{1 + \alpha}{\frac{\beta + 3}{\frac{1 + \beta}{\beta + 2}}}}{\beta + \left(\alpha + 2\right)} \]

Alternative 5
Error	1.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.85:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 3}}{\alpha + 2} \cdot \frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 4}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{1 + \alpha}{\left(-3 - \alpha\right) \cdot \left(\left(\alpha + 2\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 4}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 7
Error	1.2
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 4}}{t_0}\\ \end{array} \]

Alternative 8
Error	1.1
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(4 + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \]

Alternative 9
Error	1.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	0.8
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 120000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(4 + \alpha \cdot 2\right)}}{t_0}\\ \end{array} \]

Alternative 11
Error	1.8
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 12
Error	1.5
Cost	964

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(0.16666666666666666 + \alpha \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 4}}{t_0}\\ \end{array} \]

Alternative 13
Error	1.6
Cost	964

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 1:\\ \;\;\;\;\frac{\frac{1 + \alpha}{6 - \beta}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + 4}}{t_0}\\ \end{array} \]

Alternative 14
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.3:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 15
Error	2.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	2.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	4.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 18
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 19
Error	5.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 20
Error	34.0
Cost	320

\[\frac{0.16666666666666666}{\beta + 2} \]

Alternative 21
Error	56.7
Cost	64

\[0.1111111111111111 \]

Reproduce

herbie shell --seed 2022340 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))